In the mathematical field of functional analysis, the space denoted by c is the vector space of all convergent sequences
x
{\displaystyle \left(x_{n}\right)}
of real numbers or complex numbers.
When equipped with the uniform norm:
‖
∞
the space
becomes a Banach space.
It is a closed linear subspace of the space of bounded sequences,
ℓ
, and contains as a closed subspace the Banach space
of sequences converging to zero.
The dual of
is isometrically isomorphic to
ℓ
is reflexive.
In the first case, the isomorphism of
ℓ
ℓ
then the pairing with an element
lim
This is the Riesz representation theorem on the ordinal
ω
the pairing between
ℓ
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